Rational Functions

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					          Rational Functions
            Analysis and Graphing

Our Learning objective: Is to explore and explain why
the denominator of a rational function cannot be zero. Thus
recognizing these values as the places where vertical
asymptotes occur, (which are disastrous things to have), and
graphically what vertical asymptotes look like and mean.
    What is a Rational Function?
• A rational function has the form of a
  polynomial over a polynomial.
• The bottom polynomial must never be zero!
• If the bottom polynomial is zero this will make
  the function undefined.
• Hence, these values
  are left out of the function’s
  domain.                           Equations of
                                    rational functions
                Self Check 1
Which function below is a rational function?


   What are Vertical Asymptotes?
• Vertical asymptotes are the values that make the
  denominator go to zero, which makes the
  function undefined.
• These values are represented by x = a and/or
  x = b, where a and b are real numbers.
• What do you notice about the equations x = a
  and x = b?
• They are the equations of
   vertical lines, hence the
   name vertical asymptotes!              Basic Rational
                                              Function Shape.
 How do we find Vertical Asymptotes?
• We have to find the places where the
  denominator goes to zero.
• We do this by setting the denominator (bottom
  polynomial) equal to zero and finding the
  values of x that make it zero.
• This is when we get the
  equations of the form x = a.            Vertical line
                                              at x = a.
               Self Check 2
• What are the vertical asymptotes for the

• Answer
 Do Vertical Asymptotes Always Exist?
• No. If the zeros of the bottom polynomial are
  complex numbers, then the function does not
  have vertical asymptotes.
• See the diagram to the right?
  This rational function’s
  denominator does not go to
  zero. We can tell because the ends of the
  graph get close to the x – axis but do not cross
  the x- axis.
                Self Check 3
• Which of the following functions has vertical
  asymptotes? Set each denominator equal to
  zero and solve for the values of x.


      How can we Identify Vertical
       Asymptotes Graphically?
• Vertical asymptotes are identified as dashed
  or dotted vertical lines in the plane.
• You guessed it!! The equations of those
  vertical lines are the values of x that make the
  denominator equal to zero or x = a and x = b.
                         What do you notice about the
                         graph of the function as it
                         approaches the vertical line that
                         is the vertical asymptote?
                Self Check 4
Which diagram illustrates a vertical asymptote?
A.)                        B.)

What do Vertical Asymptotes mean to
     the graph of the function?
• Since the vertical asymptotes make the
  function undefined, the graph of the function
  NEVER crosses or touches the vertical
• Hence, the graph bends and diverges to
  positive infinity or negative infinity on each
  side of each vertical asymptote!
• See the next slide for two examples!!
     Graphs of Functions Containing
         Vertical Asymptotes.

Rational Function with one                        This rational function has
vertical asymptote. It                            two vertical asymptotes. So
diverges, bends toward                            you have to determine
positive infinity on the right                    which way it diverges on
and bends toward negative                         each side of each vertical
infinity on the left.                             asymptote. Here that is
                                                  four different calculations!!
What is the most vertical asymptotes a function
can have? The tangent function has infinite!!
               Self Check 5
Tell which way the function diverges as it
  approaches the vertical asymptote from the
  right and the left.

   Are Vertical Asymptotes Good?
• What do you think it means to say the graph bends or
  diverges toward positive or negative infinity?
• If you are analyzing the cost of producing something
  and the model’s graph bends toward positive infinity,
  do you think that is something good?
• Vertical asymptotes are disastrous things, because
  when the function diverges it means the function
  forever goes in the direction of infinity. Thus we call
  the function undefined.
• We avoid models that behave in this way because they
  are unstable and can have disastrous effects.
             Self Check Answers
1.   A.
2.   There are two: x = - 1/2 and x = 1/2
3.   B.
4.   B.
5.   From the left goes toward negative infinity
     and from the right toward positive infinity.

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